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2 Dr. Michael P. McLaughlin This tutorial is distributed free of charge and may neither be sold nor repackaged for sale in whole or in part. Macintosh is a registered trademark of Apple Computer, Inc. ii

3 PREFACE OK, I ve got some data. Now what? It is the quintessence of science, engineering, and numerous other disciplines to make quantitative observations, record them, and then try to make some sense out of the resulting dataset. Quite often, the latter is an easy task, due either to practiced familiarity with the domain or to the fact that the goals of the exercise are undemanding. However, when working at the frontiers of knowledge, this is not the case. Here, one encounters unknown territory, with maps that are sometimes poorly defined and always incomplete. The question posed above is nontrivial; the path from observation to understanding is, in general, long and arduous. There are techniques to facilitate the journey but these are seldom taught to those who need them most. My own observations, over the past twenty years, have disclosed that, if a functional relationship is nonlinear, or a probability distribution something other than Gaussian, Exponential, or Uniform, then analysts (those who are not statisticians) are usually unable to cope. As a result, approximations are made and reports delivered containing conclusions that are inaccurate and/or misleading. With scientific papers, there are always peers who are ready and willing to second-guess any published analysis. Unfortunately, there are as well many less mature disciplines which lack the checks and balances that science has developed over the centuries and which frequently address areas of public concern. These concerns lead, inevitably, to public decisions and warrant the best that mathematics and statistics have to offer, indeed, the best that analysts can provide. Since Nature is seldom linear or Gaussian, such analyses often fail to live up to expectations. The present tutorial is intended to provide an introduction to the correct analysis of data. It addresses, in an elementary way, those ideas that are important to the effort of distinguishing information from error. This distinction, unhappily not always acknowledged, constitutes the central theme of the material described herein. Both deterministic modeling (univariate regression) as well as the (stochastic) modeling of random variables are considered, with emphasis on the latter since it usually gets short shrift in standard textbooks. No attempt is made to cover every topic of relevance. Instead, attention is focussed on elucidating and illustrating core concepts as they apply to empirical data. I am a scientist, not a statistician, and these are my priorities. This tutorial is taken from the documentation included with the Macintosh software package Regress+ which is copyrighted freeware, downloadable at Michael P. McLaughlin McLean, VA October, 1999 iii

4 For deeds do die, however nobly done, And thoughts of men do as themselves decay, But wise words taught in numbers for to run, Recorded by the Muses, live for ay. E. Spenser [1591] iv

5 the very game HIS mother was a witch (or so they said). Had she not poisoned the glazier s wife with a magic drink and cursed her neighbors livestock so that they became mad? Did she not wander about the town pestering people with her recipes for medicines? Yet, in spite of such proofs, his filial devotion did not go unrewarded and, with the aid of a good lawyer plus the support of his friend and patron Rudolph II, Emperor of the Romans, King of Germany, Hungary, Bohemia, &c., Archduke of Austria, &c., the old woman made her final exit with less flamboyance than some of His Holy Imperial Majesty s subjects might have wished. However, this is not about the mother but the son and Mars. Johannes Kepler is remembered, to this day, for his insight and his vision. Even more than his contemporary, Galileo, he is honored not just for what he saw but because he invented a new way of looking. In astronomy, as in most disciplines, how you look determines what you see and, here, Kepler had a novel approach. He began with data whereas all of his predecessors had begun with circles. The Aristotelian/Ptolemaic syllogism decreed that perfect motion was circular. Heavenly bodies were perfect. Therefore, they moved in circles, however many it took to save appearances. It took a lot. When, more than a quarter of a century before Kepler, Nicolaus Copernicus finally laid down his compass, he had, quite rightly, placed the Sun in the center of the remaining seven known bodies but he had also increased the number of celestial circles to a record forty-eight! Kepler commenced his intellectual journey along the same path. Indeed, in those days, it was the only path. After many false starts, however, he realized that a collection of circles just would not work. It was the wrong model; the data demanded something else. Kepler bowed to Nature and, without apology, substituted ellipses for circles. He was the first scientist to subjugate theory to observation in a way that we would recognize and applaud. Of course, Kepler was in a unique position. Thanks to Tycho Brahe, he had the best data in the world and he was duly impressed. Still, he could have voted the party line and added yet more circles. Sooner or later, he would have accumulated enough parameters to satisfy every significant figure of every measurement. But it was wrong and it was the wrongness of it that impressed Kepler most of all. Although he drew his inspiration from the ancient Pythagoreans and the religious fervor of his own time, his words leave little doubt of his sincerity: Now, as God the maker played, he taught the game to Nature whom he created in his image: taught her the very game which he played himself Kepler searched through the data and found the game. He learned the rules and showed that, if you played well enough, sometimes even emperors take notice. Today, our motivation is different but the game goes on. We begin, of course, with data.

6 DATA There are two kinds of data: measurements and opinions. This discussion will focus exclusively on the former. In fact, although there are useful exceptions in many disciplines, here we shall discuss only quantitative measurements. Adopting this mild constraint provides two enormous advantages. The first is the advantage of being able to speak very precisely, yielding minimal concessions to the vagaries of language. The second is the opportunity to utilize the power of mathematics and, especially, of statistics. Statistics, albeit a discipline in its own right, is primarily an ever-improving cumulation of mathematical tools for extracting information from data. It is information, not data, that leads ultimately to understanding. Whenever you make measurements, perform experiments, or simply observe the Universe in action, you are collecting data. However, real data always leave something to be desired. There is an open interval of quality stretching from worthless to perfect and, somewhere in between, will be your numbers, your data. Information, on the other hand, is not permitted the luxury of imperfection. It is necessarily correct, by definition. Data are dirty; information is golden. To examine data, therefore, is to sift the silt of a riverbed in search of gold. Of course, there might not be any gold but, if there is, it will take some knowledge and considerable skill to find it and separate it from everything else. Not only must you know what gold looks like, but you also have to know what sorts of things masquerade as gold. Whatever the task, you will need to know the properties of what you seek and what you wish to avoid, the chemistry of gold and not-gold. It is through these properties one can be separated from the other. An example of real data is shown in Table 1 and Figure 1. 1 This dataset consists of values for the duration of daytime (sunrise to sunset) at Boston, Massachusetts over three years. The first day of each month has been tabulated along with the longest and shortest days occurring during this period. Daytime has been rounded off to the nearest minute. What can be said about data such as these? It can be safely assumed that they are correct to the precision indicated. In fact, Kepler s data, for analogous measurements, were much more precise. It is also clear that, at this location, the length of the day varies quite a bit during the year. This is not what one would observe near the Equator but Boston is a long way from tropical climes. Figure 1 discloses that daytime is almost perfectly repetitive from year to year, being long in the (Northern hemisphere) Summer and short in the Winter. Such qualitative remarks, however, are scarcely sufficient. Any dataset as rich as this one deserves to be further quantified in some way and, moreover, will have to be if the goal is to gain some sort of genuine understanding. With scientific data, proof of understanding implies the capability to make accurate predictions. Qualitative conclusions are, therefore, inadequate. Quantitative understanding starts with a set of well-defined metrics. There are several such metrics that may be used to summarize/characterize any set of N numbers, y i, such as these daytime values. The most common is the total-sum-of-squares, TSS, defined in Equation 1. 1 see file Examples:Daytime.in [FAM95] 2

8 Daytime HminL Day Figure 1. Raw Daytime Data N Σ Total-sum-of-squares TSS = y i y 2 i=1 1. where y is the average value of y. TSS is a positive number summarizing how much the y-values vary about their average (mean). The fact that each x (day) is paired with a unique y (daytime) is completely ignored. By discounting this important relationship, even a very large dataset may be characterized by a single number, i.e., by a statistic. The average amount of TSS attributable to each point (Equation 2) is known as the variance of the variable, y. Lastly, the square-root of the variance is the standard deviation, another important statistic. Variance of y Var y = 1 N N Σ i=1 y i y 2 2. In Figure 1, the y-values come from a continuum but the x-values do not. More often, x is a continuous variable, sampled at points chosen by the observer. For this reason, it is called the independent variable. The dependent variable, y, describes measurements made at chosen values of x and is almost always inaccurate to some degree. Since the x-values are selected in advance by the observer, they are most often assumed to be known exactly. Obviously, this cannot be true if x is a real number but, usually, uncertainties in x are negligible compared to uncertainties in y. When this is not true, some very subtle complications arise. 4

9 Table 2 lists data from a recent astrophysics experiment, with measurement uncertainties explicitly recorded. 2 These data come from observations, made in , of comet Hale- Bopp as it approached the Sun [RAU97]. Here, the independent variable is the distance of the comet from the Sun. The unit is AU, the average distance (approximately) of the Earth from the Sun. The dependent variable is the rate of production of cyanide, CN, a decomposition product of hydrogen cyanide, HCN, with units of molecules per second divided by Thus, even when Hale-Bopp was well beyond the orbit of Jupiter (5.2 AU), it was producing cyanide at a rate of (6 ± 3) x 1025 molecules per second, that is, nearly 2.6 kg/s. Table 2. Rate of Production of CN in Comet Hale-Bopp Rate Distance from Sun Uncertainty in Rate (molecules per second)/1025 (AU) (molecules per second)/ In this example, the uncertainties in the measurements (Table 2, column 3) are a significant fraction of the observations themselves. Establishing the value of the uncertainty for each data point and assessing the net effect of uncertainties are crucial steps in any analysis. Had Kepler s data been as poor as the data available to Copernicus, his name would be known only to historians. The data of Table 2 are presented graphically in Figure 2. For each point, the length of the error bar indicates the uncertainty 3 in y. These uncertainties vary considerably and with some regularity. Here, as often happens with observations made by electronic instruments which measure a physical quantity proportional to the target variable, the uncertainty in an observation tends to increase with the magnitude of the observed value. Qualitatively, these data suggest the hypothesis that the comet produced more and more CN as it got closer to the Sun. This would make sense since all chemical reactions go faster as the temperature increases. On the other hand, the observed rate at 2.9 AU seems too small. Did the comet simply start running out of HCN? How likely is it that the rate at 3.1 AU was really bigger than the rate at 2.9 AU? Are these values correct? Are the uncertainties correct? If the uncertainties are correct, what does this say about the validity of the hypothesis? All of these are legitimate questions. 2 see file Examples:Hale_Bopp.CN.in 3 In spite of its name, this bar does not indicate error. If it did, the error could be readily removed. 5

10 Rate Distance HAUL Figure 2. Hale-Bopp CN Data Finally, consider the very unscientific data shown in Figure 3. This figure is a plot of the highest major-league baseball batting averages in the United States, for the years , as a function of time. 4 A player s batting average is the fraction of his official at-bats in which he hit safely. Thus, it varies continuously from zero to one. It is fairly clear that there is a large difference between these data and those shown in Figure 1. The latter look like something from a math textbook. One gets the feeling that a daytime value could be predicted rather well from the values of its two nearest neighbors. There is no such feeling regarding the data in Figure 3. At best, it might be said that batting champions did better before World War II than afterwards. However, this is not an impressive conclusion given nearly a hundred data points. Considering the data in Figure 3, there can be little doubt that maximum batting average is not really a function of time. Indeed, it is not a function of anything. It is a random variable and its values are called random variates, a term signifying no pretense whatever that any of these values are individually predictable. 5 When discussing random (stochastic) variables, the terms independent and dependent have no relevance and are not used, nor are scatter plots such as Figure 3 ever drawn except to illustrate that they are almost meaningless. 4 see files Examples:BattingAvgEq.in and Examples:BattingAvg.in [FAM98] 5 The qualification is crucial; it makes random data comprehensible. 6

11 0.45 Batting Average Year Figure 3. Annual Best Baseball Batting Average Variables appear random for one of two reasons. Either they are inherently unpredictable, in principle, or they simply appear so to an observer who happens to be missing some vital information that would render them deterministic (non-random). Although deterministic processes are understandably of primary interest, random variables are actually the more common simply because that is the nature of the Universe. In fact, as the next section will describe in detail, understanding random variables is an essential prerequisite for understanding any real dataset. Making sense of randomness is not the paradox it seems. Some of the metrics that apply to deterministic variables apply equally well to random variables. For instance, the mean and variance (or standard deviation) of these batting averages could be computed just as easily as with the daytime values in Example 1. A computer would not care where the numbers came from. No surprise then that statistical methodology may be profitably applied in both cases. Even random data have a story to tell. Which brings us back to the point of this discussion. We have data; we seek insight and understanding. How do we go from one to the other? What s the connection? The answer to this question was Kepler s most important discovery. Data are connected to understanding by a model. When the data are quantitative, the model is a mathematical model, in which case, not only does the form of the model lead directly to understanding but one may query the model to gain further information and insight. But, first, one must have a model. 7

12 FROM DATA TO MODEL Marshall McLuhan is widely recognized for his oft-quoted aphorism, The medium is the message. It is certainly true in the present context. The model is the medium between data and understanding and it is, as well, the message of this tutorial. The current section presents an outline of fundamental concepts essential for initiating the first step, from data to model. The methodology for implementing these concepts is described in later sections. The second step, from model to understanding, will be left as an exercise for the reader. As can be seen, through the examples above, data are not transparent. They do not reveal their secrets to casual observers. Yet, when acquired with due diligence, they contain useful information. The operative word is contain. Data and information are not equivalent even though, in colloquial speech, they are treated as such. Why not? What is there in a dataset that is not information? The answer is error. Data = Information + Error Were it not for error, every observation would be accurate, every experiment a paragon of perfection, and nearly every high school Science Fair project a fast track to the Nobel Prize. Alas, error exists. It exists not just in every dataset but in every data point in every dataset. It contaminates everything it touches and it touches everything. To make any progress, it must be identified and filtered out. A good model does this very nicely. In its simplest, most superficial aspect, a model is a filter designed to separate data into these two components. A mathematical model is a statistical filter that not only attempts the separation but also quantifies its success in that effort. Two Ways to Construct a Model To construct a model, it is necessary to proceed from the known to the unknown or, at the very least, from the better known to the less well known. There are two approaches. The choice depends upon whether it is the information or the error that is better known, bearing in mind that known and assumed are not synonyms. In the first case, the model is designed to utilize the known properties of the embedded information to extract the latter and leave the error behind. This approach is commonly employed with stochastic data. Alternatively, if the error is the better known, the model is designed to operate on the error, filtering it out and leaving the information behind. This approach is nearly universal with deterministic data. In either case, the separation will be imperfect and the information still a bit erroneous. The notion of deterministic information is commonplace and requires no further elaboration but what about the putative stochastic information contained in a dataset of random variables? Is it real? Likewise, can anything useful be said about error? Do random variables and error really have properties that one can understand? In other words, are they comprehensible? They are indeed. The remainder of this section elucidates some properties of i) stochastic information and ii) error. Such properties are quantitative, leading directly to the identification of optimization criteria which are crucial to any modeling process. 8

13 Stochastic Information Stochastic. Information. The juxtaposition seems almost oxymoronic. If something is stochastic (random), does that not imply the absence of information? Can accurate conclusions really be induced from random variables? Well, yes, as a matter of fact. That a variable is stochastic signifies only that its next value is unpredictable, no matter how much one might know about its history. The data shown in Figure 3 are stochastic in this sense. If you knew the maximum batting average for every year except 1950, that still would not be enough information to tell you the correct value for the missing year. In fact, no amount of ancillary data would be sufficient. This quality is, for all practical purposes, the quintessence of randomness. Yet, we have a very practical purpose here. We want to assert something definitive about stochastic information. We want to construct models for randomness. Such models are feasible. Although future values for a random variable are unpredictable in principle, their collective behavior is not. Were it missing from Figure 3, the maximum batting average for 1950 could not be computed using any algorithm. However, suggested values could be considered and assessed with respect to their probability. For instance, one could state with confidence that the missing value is probably not less than or greater than One could be quite definite about the improbability of an infinite number of candidate values because any large collection of random variates will exhibit consistency in spite of the randomness of individual variates. It is a matter of experience that the Universe is consistent even when it is random. A mathematical model for randomness is expressed as a probability density function (PDF). The easiest way to understand a PDF is to consider the process of computing a weighted average. For example, what is the expectation (average value) resulting from tossing a pair of ordinary dice if prime totals are discounted? There are six possible outcomes but they are not equally probable. Therefore, to get the correct answer, one must calculate the weighted average instead of just adding up the six values and dividing the total by six. This requires a set of weights, as shown in Table 3. Table 3. Weights for Non-prime Totals for Two Dice Value Weight 4 3/21 6 5/21 8 5/21 9 4/ / /21 The random total, y, has an expected value, denoted <y> or y, given by Equation 3. 9

14 6 Σ Expectation of y y = w i y i i=1 3. The true expectation, 7.62, is quite different from the unweighted average, The set of weights in Table 3 constitutes a discrete PDF, meaning that there are a countable number of possible values, each with its own weight (density). Given a discrete PDF, f(y), for the random variable, y, any arbitrary function of y, g(y), will have an expectation computed as shown in Equation 4. Equation 3 is just a special case of Equation 4. g(y) = f(y i )g(y i ) y Σall i 4. where y is equal to the binwidth, defined below in Example S1. In the continuous case, neighboring values are separated by the infinitesimal binwidth, dy. Nevertheless, a continuous PDF has exactly the same meaning as a discrete PDF and, thus, the continuous analogue of Equation 4 is Equation 5. g(y) = f(y) g(y) dy 5. For example, suppose f(y) = e y and y 0. Then, the expectation of g(y) = y is y = e y y dy 0 = π 2 6. The product (f(y) y) or (f(y) dy) equals the probability of a given y. Probability is a dimensionless quantity. That is, it is a pure number with no units. Consequently, the units of a PDF must always be the reciprocal of the units, if any, of y. 6 It is for this reason that the function f(y) is referred to as a density function. Example S1 Batting Averages The dataset shown in Figure 3 will be the first example. The collective behavior of these data can be summarized numerically and graphically. For instance, their mean (expectation) is and their variance is [st. dev. = ]. Any statistics textbook will list several additional metrics that could be used to characterize this set of 97 values. For a graphical depiction, the device most commonly employed is the histogram, constructed by grouping the variates into numerical intervals of equal width and plotting these bins against their frequencies, the number of variates in the respective bins. These frequencies may be converted to probabilities (normalized) by dividing each by the sample size and then converted to probability densities by further dividing by the binwidth (here, 0.011). When all of this is done with the batting-average data, we get the histogram shown in Figure 4 (gray boxes). 6 a useful check for complicated density formulas (see Appendix A) 10

15 Figure 4. Batting-Average Data (Histogram and PDF) The overall shape of a histogram is intended to describe how realizable values of a random variable are scattered across the infinite landscape of conceivable locations. The way it works is that the histogram is constructed to have a total area = 1. Therefore, the probability that any random variate falls in a given bin is equal to the area of the corresponding rectangle. This scattering of values/probabilities is referred to as the distribution of the random variable. Note that the Y-axis measures probability density, not probability. Given the computational process just outlined, and recalling Equations 4 and 5, measurements along this axis have units of 1/u, where u is the unit of the random variable (e.g., inches, grams, etc.) A continuous density function for these data is also shown in Figure 4 (solid line). The corresponding analytical expression, given in Equation 7, is called the Gumbel distribution. 7 Like the histogram, the total area under this curve, including the tails (not shown), equals 1. PDF = 1 B exp A y B exp exp A y B 7. where, for this example, A = and B = see pg. A-49 11

16 We see that Equation 7 does have the proper units. Parameter A has the same units as the random variable, y. Since any exponent is necessarily dimensionless, parameter B must have the same units as well. Hence, the entire expression has units of 1/u. Of course, these particular data happen to be dimensionless already, but variates described by a PDF usually do have units of some sort. The Gumbel distribution is one example of a continuous distribution. [There are many others described in Appendix A.] If f(y) is the PDF, the probability of any value of y would equal f(y)*dy which, since dy is vanishingly small, is zero. Indeed, intuition demands that the probability of randomly picking any given value from a continuum must be zero. However, the probability of picking a value in a finite range [a, b] may be greater than zero (Equation 8). b Prob a y b = f y dy a 8. The integral (or sum) of a density function, from minus infinity to some chosen value, y, is referred to as the cumulative distribution function (CDF), usually denoted F(y). 8 For this example, it is plotted in Figure 5. In this figure, the solid line is the theoretical CDF and the gray line the empirical CDF. The CDF-axis gives the probability that a randomly selected variate will be less than or equal to b (Equation 9). b Prob y b =Fb = f y dy 9. Some of the reasons for choosing the Gumbel distribution for these data are discussed in its entry in Appendix A. For now, it is sufficient to appreciate that any sort of theoretical expression would be considered relevant. This supports the conclusion that useful statements can be made about stochastic data. However, it does not explain where these parameter values came from. They are another matter entirely. One of the reasons for using the values of A and B given above is that they produce a curve that encloses an area with approximately the same shape and size as the empirical histogram. There are much better reasons but their explication requires some preliminary concepts. The first is the likelihood of a dataset. The term means just what it says. It indicates how likely this set of observations would be had they been selected randomly from the given PDF. If the variates are independent, the likelihood is also a measure of their joint probability. 9 For any PDF, f(y), the likelihood of a sample of N variates is defined by Equation 10. Likelihood N Πk=1 f y k The term distribution is sometimes taken to mean the cumulative distribution. 9 Two random quantities, x and xx, are independent if and only if their joint (simultaneous) probability, Prob(x AND xx), is always equal to Prob(x)*Prob(xx). 12

17 Figure 5. Batting-Average Data (CDF) The second concept is that of maximum likelihood (ML). This notion is also very intuitive. Given a form for the density function, the ML parameters are those parameters which make the likelihood as large as possible. In other words, any other set of parameters would make the observed dataset less likely to have been observed. Since the dataset was observed, the ML parameters are generally considered to be optimal. 10 However, this does not preclude the use of some alternate criterion should circumstances warrant. The values given above for A and B are the ML parameters. This is an excellent reason for choosing them to represent the data and it is no coincidence that the theoretical curve in Figure 4 matches the histogram so well. Had there been a thousand data points, with proportionately narrower histogram bins, the match would have been even better. Thus, a Gumbel distribution, with these ML parameters, is a model for the data shown in Figure 3. It summarizes and characterizes the stochastic information contained in these data. Not only does it describe the probability density of the data, but one could query the model. One could ask the same questions of the model that could be asked of the data. For example, What are the average and standard deviation of these data? or What is the probability that next year s batting champion will have a batting average within five percent of the historical 10 Their uniqueness is usually taken for granted. 13

18 mean? The answers to such questions may be computed from the model without looking at the data at all. For instance, the last question is answered in Equation Probability = 1.05 * mean 0.95 * mean f y dy = F F = As a check, note that Equation 11 predicts that 53 of the values in Figure 3 are within the given range. The data show 52 in this range. One could even ask questions of the model that one could not ask of the data. For instance, What is the probability that next year s batting champion will have a batting average greater than the maximum in the dataset (0.424)? Obviously, there is no such number in the historical data and that suggests an answer of zero. However, this is clearly incorrect. Sooner or later, someone is bound to beat provided that major-league baseball continues to be played. It is easy to pose a theoretical question like this to a model and, if the model is good, to obtain a correct answer. This model (Equation 7) gives an answer of three percent, suggesting that we are already overdue for such a feat. Whether or not a model is good (valid) is a question yet to be addressed. Example S2 Rolling Dice The Gumbel distribution was chosen to model the batting-average data because it is known to be appropriate, in many cases, for samples of continuous-distribution maxima (so-called record values). However, there is no a priori guarantee that it is valid for these particular data. Occasionally, there is enough known about some type of stochastic information that theory alone is sufficient to specify one distribution to the exclusion of all others and, perhaps, even specify the parameters as well. The second stochastic example illustrates this situation via the following experiment. Step 1 Think of a number, a whole number, from one to six. Step 2 Roll a standard, cubical die repeatedly until your chosen number appears three times. Step 3 Record the number of rolls, Nr, required. After only a few semesters of advanced mathematics courses, it would be relatively easy to prove that the random variable, N r, is ~NegativeBinomial(1/6, 3). 11 This assertion could be tested by performing the experiment a very large number of times and comparing the theoretical distribution to the empirical data. 11 see pg. A-81; the ~ is read (is) distributed as 14

19 Carrying out this exercise by hand would be a bit tedious. Fortunately, it is a simple matter to simulate the experiment on a computer. Such simulations are very common. Simulated results for a 1,000-fold repetition of this experiment are shown in Figure Figure 6. Rolls3 Data (Histogram and PDF) The NegativeBinomial model shown in Figure 6 (solid lines) is another example of a discrete PDF. Typically, this means that the random variable can take on only integer values. That is the case here since N r is a count of the number of rolls required. In computing the model shown, the first parameter was not fixed at its theoretical value, 1/6. Instead, it was estimated from the data. The ML estimate = , very close to the theoretical It would have been even closer had the sample size been much larger than 1,000. Likewise, the mean predicted by this model = 17.74, very close to the theoretical mean, 18. The second parameter, 3, was assumed given although it, too, could have been considered unknown and estimated from the data. The observed data (gray histogram) may be compared to the estimated, ML model using the Chi-square statistic, defined in Equation see file Examples:Rolls3.in 15

20 Chi square χ 2 = Σall k o k e k 2 e k 12. where k includes all possible values of the random variable and where o and e are the observed and expected frequencies, respectively. For this example, Chi-square = Again, the question of whether this amount of discrepancy between theory and experiment is acceptable is a matter to be discussed below. Example S3 Normality We would be remiss in our illustrations of stochastic information were we to omit the most famous of all stochastic models, the Normal (Gaussian) distribution. The Normal distribution is a continuous distribution and it arises naturally as a result of the Central Limit Theorem. In plain English, this theorem states that, provided they exist, averages (means) of any random variable or combination of random variables tend to be ~Normal(A, B), where A is the sample mean and B is the unbiased estimate of the population standard deviation. As usual, with random data, the phrase tend to indicates that the validity of this Gaussian model for random means increases as the sample size increases. In Example S2, a thousand experiments were tabulated and individual outcomes recorded. A sample size of 1,000 is certainly large enough to illustrate the Central Limit Theorem. Our final example will, therefore, be a 1,000-fold replication of Example S2, with one alteration. Instead of recording 1,000 values of N r for each replicate, only their average, N avg, will be recorded. This new experiment will thus provide 1,000 averages 13 which, according to the Central Limit Theorem, should be normally distributed. As before, the experiment was carried out in simulation. The observed results and the ML model are shown in Figure 7. The estimated values for A and B are and , respectively. With a Gaussian model, the ML parameter estimates may be computed directly from the data so one cannot ask how well they match the observed values. A somewhat better test of the model would be to pick a statistic not formally included in the analytical form of the distribution. One such statistic is the interquartile range, that is, the range included by the middle half of the sorted data. In this example, the model predicts a range of [17.79, 18.19] 14 while the data show a range of [17.81, 18.19]. The ideal test would be one that was independent of the exact form of the density function. For continuous distributions, the most common such test is the Kolmogorov-Smirnov (K-S) statistic. The K-S statistic is computed by comparing the empirical CDF to the theoretical CDF. For example S3, these two are shown in Figure 8. The K-S statistic is simply the magnitude (absolute value) of the maximum discrepancy between the two, measured along the CDF-axis. Here, the K-S statistic = see file Examples:Rolls3avg.in 14 see pg. A-85 16

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